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In the normal metric for spacetime $ds^2=dx^2+dy^2+dz^2-cdt^2$ What does a "circle" of unit distance look like (the set of all points that satisfy $ds^2=1$?) That is does the negative give this any unusual properties as compared to the usual 4d sphere in euclidean space? What would the "$\pi$" be in this metric?

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    $\begingroup$ Instead of a 4D sphere you get a 4D hyperboloid. $\endgroup$ – David K Jul 12 at 12:45
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It is a $4D$ hyperboloid which is preserved by its associated symmetries: rotations along the $3D$ space axis, as well as Lorenz transformations along the time axis.

Lorenz transformations are the "hyperbolic equivalent" of rotations. They're what happens when one axis is space-like and one is time-like. So the distance function is $dc^2 = dx^2 - dy^2$ instead of the usual Pythagorean one .The rules for Lorenz transformations is they preserve light-cones (From a physics perspective, something moving at light-speed in one frame will be seen as moving at light speed in any other valid frame. From a mathematics perspective, points on a light-cone have $0 = dx^2 - dx^2 $ distance between them, and Lorenz transformation, being analogous to rotation, is required to preserve said distances), as well as the total "volume" of a given region of space-time. So Lorenz transformations look kind of like area-preserving "stretches" along the light-cones as axis:

https://upload.wikimedia.org/wikipedia/commons/b/b4/Animated_Lorentz_Transformation.gif

The figures that are preserved by such transformations are those hyperbola "centered" at (0,0) . In the same way that the figures preserved by rotation around (0,0) are the circles centered around it. It can be generalized to an arbitrary number of space-like and time-like axis. $dc^2 = dx^2 + dy^2 + dz^2 - dt^2$.

An intuitive introduction to hyperbolic geometry (what happens when transformations preserve the distance given by $dc^2 = dx^2 - dy^2$ instead of $dc^2 = dx^2 + dy^2$) is presented here:

http://www.dynamicgeometry.com/Documents/advancedSketchGallery/minkowski/Minkowski_Overview.pdf

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